\(\int \frac {\cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx\) [431]

Optimal result

Integrand size = 21, antiderivative size = 188 \[ \int \frac {\cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}-\frac {2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^6 d}+\frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d} \]

[Out]

1/8*a*(8*a^4-20*a^2*b^2+15*b^4)*x/b^6-2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^6/d
+1/5*cos(d*x+c)^5/b/d-1/12*cos(d*x+c)^3*(4*a^2-4*b^2-3*a*b*sin(d*x+c))/b^3/d+1/8*cos(d*x+c)*(8*(a^2-b^2)^2-a*b
*(4*a^2-7*b^2)*sin(d*x+c))/b^5/d

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2774, 2944, 2814, 2739, 632, 210} \[ \int \frac {\cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^6 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {a x \left (8 a^4-20 a^2 b^2+15 b^4\right )}{8 b^6}+\frac {\cos ^5(c+d x)}{5 b d} \]

[In]

Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x]),x]

[Out]

(a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*x)/(8*b^6) - (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 -
b^2]])/(b^6*d) + Cos[c + d*x]^5/(5*b*d) - (Cos[c + d*x]^3*(4*(a^2 - b^2) - 3*a*b*Sin[c + d*x]))/(12*b^3*d) + (
Cos[c + d*x]*(8*(a^2 - b^2)^2 - a*b*(4*a^2 - 7*b^2)*Sin[c + d*x]))/(8*b^5*d)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2774

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + p))), x] + Dist[g^2*((p - 1)/(b*(m + p))), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] &&
NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2944

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) -
a*d*p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(m + p)*(m + p +
1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1
) - d*(a^2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2,
0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*m]

Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^5(c+d x)}{5 b d}+\frac {\int \frac {\cos ^4(c+d x) (b+a \sin (c+d x))}{a+b \sin (c+d x)} \, dx}{b} \\ & = \frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\int \frac {\cos ^2(c+d x) \left (-b \left (a^2-4 b^2\right )-a \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 b^3} \\ & = \frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}+\frac {\int \frac {b \left (4 a^4-9 a^2 b^2+8 b^4\right )+a \left (8 a^4-20 a^2 b^2+15 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{8 b^5} \\ & = \frac {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}-\frac {\left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^6} \\ & = \frac {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}-\frac {\left (2 \left (a^2-b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^6 d} \\ & = \frac {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}+\frac {\left (4 \left (a^2-b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^6 d} \\ & = \frac {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}-\frac {2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^6 d}+\frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(507\) vs. \(2(188)=376\).

Time = 5.33 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.70 \[ \int \frac {\cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\cos (c+d x) \left (240 (a-b)^3 (a+b)^2 \text {arctanh}\left (\frac {\sqrt {a-b} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{\sqrt {a+b} \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {a+b} \left (-240 \sqrt {a-b} \left (a^2-b^2\right )^2 \text {arctanh}\left (\frac {\sqrt {\frac {b (1+\sin (c+d x))}{-a+b}}}{\sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \left (30 \sqrt {b} \left (8 a^4-4 a^3 b-16 a^2 b^2+7 a b^3+8 b^4\right ) \text {arcsinh}\left (\frac {\sqrt {a-b} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{\sqrt {2} \sqrt {b}}\right )+\sqrt {a-b} \sqrt {1-\sin (c+d x)} \sqrt {\frac {b (1+\sin (c+d x))}{-a+b}} \left (8 \left (15 a^4-35 a^2 b^2+23 b^4\right )-15 a b \left (4 a^2-9 b^2\right ) \sin (c+d x)+8 b^2 \left (5 a^2-11 b^2\right ) \sin ^2(c+d x)-30 a b^3 \sin ^3(c+d x)+24 b^4 \sin ^4(c+d x)\right )\right )\right )\right )}{120 \sqrt {a-b} b^5 \sqrt {a+b} d \sqrt {1-\sin (c+d x)} \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}} \]

[In]

Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x]),x]

[Out]

(Cos[c + d*x]*(240*(a - b)^3*(a + b)^2*ArcTanh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[a +
b]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))])]*Sqrt[1 - Sin[c + d*x]] + Sqrt[a + b]*(-240*Sqrt[a - b]*(a^2 - b
^2)^2*ArcTanh[Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]/Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]]*Sqrt[1 - Sin[c +
d*x]] + Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*(30*Sqrt[b]*(8*a^4 - 4*a^3*b - 16*a^2*b^2 + 7*a*b^3 + 8*b^4)
*ArcSinh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[2]*Sqrt[b])] + Sqrt[a - b]*Sqrt[1 - Sin[c
+ d*x]]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(8*(15*a^4 - 35*a^2*b^2 + 23*b^4) - 15*a*b*(4*a^2 - 9*b^2)*Sin[
c + d*x] + 8*b^2*(5*a^2 - 11*b^2)*Sin[c + d*x]^2 - 30*a*b^3*Sin[c + d*x]^3 + 24*b^4*Sin[c + d*x]^4)))))/(120*S
qrt[a - b]*b^5*Sqrt[a + b]*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Si
n[c + d*x]))/(a - b))])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(381\) vs. \(2(174)=348\).

Time = 1.28 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.03

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{3} b^{2}-\frac {9}{8} a \,b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (b \,a^{4}-3 b^{3} a^{2}+3 b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{3} b^{2}-\frac {5}{4} a \,b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (4 b \,a^{4}-10 b^{3} a^{2}+6 b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 b \,a^{4}-\frac {40}{3} b^{3} a^{2}+\frac {28}{3} b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{3} b^{2}+\frac {5}{4} a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (4 b \,a^{4}-\frac {26}{3} b^{3} a^{2}+\frac {14}{3} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{3} b^{2}+\frac {9}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+b \,a^{4}-\frac {7 b^{3} a^{2}}{3}+\frac {23 b^{5}}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {a \left (8 a^{4}-20 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{6}}+\frac {2 \left (-a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}+b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{6} \sqrt {a^{2}-b^{2}}}}{d}\)	\(382\)
default	\(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{3} b^{2}-\frac {9}{8} a \,b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (b \,a^{4}-3 b^{3} a^{2}+3 b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{3} b^{2}-\frac {5}{4} a \,b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (4 b \,a^{4}-10 b^{3} a^{2}+6 b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 b \,a^{4}-\frac {40}{3} b^{3} a^{2}+\frac {28}{3} b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{3} b^{2}+\frac {5}{4} a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (4 b \,a^{4}-\frac {26}{3} b^{3} a^{2}+\frac {14}{3} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{3} b^{2}+\frac {9}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+b \,a^{4}-\frac {7 b^{3} a^{2}}{3}+\frac {23 b^{5}}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {a \left (8 a^{4}-20 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{6}}+\frac {2 \left (-a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}+b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{6} \sqrt {a^{2}-b^{2}}}}{d}\)	\(382\)
risch	\(\frac {a^{5} x}{b^{6}}-\frac {5 a^{3} x}{2 b^{4}}+\frac {15 a x}{8 b^{2}}+\frac {{\mathrm e}^{i \left (d x +c \right )} a^{4}}{2 b^{5} d}-\frac {9 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{8 b^{3} d}+\frac {11 \,{\mathrm e}^{i \left (d x +c \right )}}{16 b d}+\frac {{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{2 b^{5} d}-\frac {9 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{8 b^{3} d}+\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 b d}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{4}}{d \,b^{6}}-\frac {2 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{2}}{d \,b^{4}}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{2}}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{4}}{d \,b^{6}}+\frac {2 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) a^{2}}{d \,b^{4}}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{2}}+\frac {\cos \left (5 d x +5 c \right )}{80 b d}+\frac {a \sin \left (4 d x +4 c \right )}{32 b^{2} d}-\frac {\cos \left (3 d x +3 c \right ) a^{2}}{12 b^{3} d}+\frac {7 \cos \left (3 d x +3 c \right )}{48 b d}-\frac {a^{3} \sin \left (2 d x +2 c \right )}{4 b^{4} d}+\frac {a \sin \left (2 d x +2 c \right )}{2 b^{2} d}\)	\(563\)

[In]

int(cos(d*x+c)^6/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(2/b^6*(((1/2*a^3*b^2-9/8*a*b^4)*tan(1/2*d*x+1/2*c)^9+(a^4*b-3*a^2*b^3+3*b^5)*tan(1/2*d*x+1/2*c)^8+(a^3*b^
2-5/4*a*b^4)*tan(1/2*d*x+1/2*c)^7+(4*a^4*b-10*a^2*b^3+6*b^5)*tan(1/2*d*x+1/2*c)^6+(6*b*a^4-40/3*b^3*a^2+28/3*b
^5)*tan(1/2*d*x+1/2*c)^4+(-a^3*b^2+5/4*a*b^4)*tan(1/2*d*x+1/2*c)^3+(4*b*a^4-26/3*b^3*a^2+14/3*b^5)*tan(1/2*d*x
+1/2*c)^2+(-1/2*a^3*b^2+9/8*a*b^4)*tan(1/2*d*x+1/2*c)+b*a^4-7/3*b^3*a^2+23/15*b^5)/(1+tan(1/2*d*x+1/2*c)^2)^5+
1/8*a*(8*a^4-20*a^2*b^2+15*b^4)*arctan(tan(1/2*d*x+1/2*c)))+2*(-a^6+3*a^4*b^2-3*a^2*b^4+b^6)/b^6/(a^2-b^2)^(1/
2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.57 \[ \int \frac {\cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\left [\frac {24 \, b^{5} \cos \left (d x + c\right )^{5} - 40 \, {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 60 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 120 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right ) + 15 \, {\left (2 \, a b^{4} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, b^{6} d}, \frac {24 \, b^{5} \cos \left (d x + c\right )^{5} - 40 \, {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 120 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 120 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right ) + 15 \, {\left (2 \, a b^{4} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, b^{6} d}\right ] \]

[In]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

[1/120*(24*b^5*cos(d*x + c)^5 - 40*(a^2*b^3 - b^5)*cos(d*x + c)^3 + 15*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*d*x + 6
0*(a^4 - 2*a^2*b^2 + b^4)*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2
+ 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c)
- a^2 - b^2)) + 120*(a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c) + 15*(2*a*b^4*cos(d*x + c)^3 - (4*a^3*b^2 - 7*a*b^4
)*cos(d*x + c))*sin(d*x + c))/(b^6*d), 1/120*(24*b^5*cos(d*x + c)^5 - 40*(a^2*b^3 - b^5)*cos(d*x + c)^3 + 15*(
8*a^5 - 20*a^3*b^2 + 15*a*b^4)*d*x + 120*(a^4 - 2*a^2*b^2 + b^4)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/
(sqrt(a^2 - b^2)*cos(d*x + c))) + 120*(a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c) + 15*(2*a*b^4*cos(d*x + c)^3 - (4
*a^3*b^2 - 7*a*b^4)*cos(d*x + c))*sin(d*x + c))/(b^6*d)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6/(a+b*sin(d*x+c)),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (173) = 346\).

Time = 0.53 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.64 \[ \int \frac {\cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {15 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} {\left (d x + c\right )}}{b^{6}} - \frac {240 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{6}} + \frac {2 \, {\left (60 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 360 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 360 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 480 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1200 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 720 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 720 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1600 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1120 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 480 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1040 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 560 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, a^{4} - 280 \, a^{2} b^{2} + 184 \, b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} b^{5}}}{120 \, d} \]

[In]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/120*(15*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*(d*x + c)/b^6 - 240*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(pi*floor(1/
2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^6) + 2
*(60*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 135*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*a^4*tan(1/2*d*x + 1/2*c)^8 - 360*a^
2*b^2*tan(1/2*d*x + 1/2*c)^8 + 360*b^4*tan(1/2*d*x + 1/2*c)^8 + 120*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 150*a*b^3*t
an(1/2*d*x + 1/2*c)^7 + 480*a^4*tan(1/2*d*x + 1/2*c)^6 - 1200*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 + 720*b^4*tan(1/2
*d*x + 1/2*c)^6 + 720*a^4*tan(1/2*d*x + 1/2*c)^4 - 1600*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 1120*b^4*tan(1/2*d*x
+ 1/2*c)^4 - 120*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 150*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 480*a^4*tan(1/2*d*x + 1/2*c
)^2 - 1040*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 560*b^4*tan(1/2*d*x + 1/2*c)^2 - 60*a^3*b*tan(1/2*d*x + 1/2*c) + 1
35*a*b^3*tan(1/2*d*x + 1/2*c) + 120*a^4 - 280*a^2*b^2 + 184*b^4)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*b^5))/d

Mupad [B] (verification not implemented)

Time = 7.23 (sec) , antiderivative size = 3075, normalized size of antiderivative = 16.36 \[ \int \frac {\cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^6/(a + b*sin(c + d*x)),x)

[Out]

((2*(15*a^4 + 23*b^4 - 35*a^2*b^2))/(15*b^5) + (tan(c/2 + (d*x)/2)^3*(5*a*b^2 - 4*a^3))/(2*b^4) - (tan(c/2 + (
d*x)/2)^7*(5*a*b^2 - 4*a^3))/(2*b^4) - (tan(c/2 + (d*x)/2)^9*(9*a*b^2 - 4*a^3))/(4*b^4) + (2*tan(c/2 + (d*x)/2
)^8*(a^4 + 3*b^4 - 3*a^2*b^2))/b^5 + (4*tan(c/2 + (d*x)/2)^6*(2*a^4 + 3*b^4 - 5*a^2*b^2))/b^5 + (4*tan(c/2 + (
d*x)/2)^2*(6*a^4 + 7*b^4 - 13*a^2*b^2))/(3*b^5) + (4*tan(c/2 + (d*x)/2)^4*(9*a^4 + 14*b^4 - 20*a^2*b^2))/(3*b^
5) + (tan(c/2 + (d*x)/2)*(9*a*b^2 - 4*a^3))/(4*b^4))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10
*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) + (atan((((-(a + b)^5*(a - b)^5)^
(1/2)*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)
*(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b
^15) + ((-(a + b)^5*(a - b)^5)^(1/2)*((28*a^2*b^16 - 44*a^4*b^14 + 16*a^6*b^12)/b^14 + ((-(a + b)^5*(a - b)^5)
^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15)))/b^6 - (tan(c/2 + (d*x)/2)*(12
8*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15)))/b^6)*1i)/b^6 + ((-(a + b)^5*(a - b)^5)^(1/2
)*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*(64
*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15)
+ ((-(a + b)^5*(a - b)^5)^(1/2)*(((-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19
- 128*a^3*b^17))/(2*b^15)))/b^6 - (28*a^2*b^16 - 44*a^4*b^14 + 16*a^6*b^12)/b^14 + (tan(c/2 + (d*x)/2)*(128*a*
b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15)))/b^6)*1i)/b^6)/((32*a^16 - 120*a^2*b^14 + 655*a^
4*b^12 - 1549*a^6*b^10 + 2069*a^8*b^8 - 1695*a^10*b^6 + 856*a^12*b^4 - 248*a^14*b^2)/b^14 + ((-(a + b)^5*(a -
b)^5)^(1/2)*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d
*x)/2)*(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5)
)/(2*b^15) + ((-(a + b)^5*(a - b)^5)^(1/2)*((28*a^2*b^16 - 44*a^4*b^14 + 16*a^6*b^12)/b^14 + ((-(a + b)^5*(a -
b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15)))/b^6 - (tan(c/2 + (d*x)/
2)*(128*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15)))/b^6))/b^6 - ((-(a + b)^5*(a - b)^5)^(
1/2)*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*
(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^
15) + ((-(a + b)^5*(a - b)^5)^(1/2)*(((-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^
19 - 128*a^3*b^17))/(2*b^15)))/b^6 - (28*a^2*b^16 - 44*a^4*b^14 + 16*a^6*b^12)/b^14 + (tan(c/2 + (d*x)/2)*(128
*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15)))/b^6))/b^6 + (tan(c/2 + (d*x)/2)*(128*a^17 -
450*a^3*b^14 + 2550*a^5*b^12 - 6230*a^7*b^10 + 8530*a^9*b^8 - 7088*a^11*b^6 + 3584*a^13*b^4 - 1024*a^15*b^2))/
b^15))*(-(a + b)^5*(a - b)^5)^(1/2)*2i)/(b^6*d) + (a*atan(((a*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9
- 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*
b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) + (a*(8*a^4 + 15*b^4 - 20*a^2*b^2)*((28*a^2*b^16
- 44*a^4*b^14 + 16*a^6*b^12)/b^14 - (tan(c/2 + (d*x)/2)*(128*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^
12))/(2*b^15) + (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15))*(8*a^4 + 15*b^4 -
20*a^2*b^2)*1i)/(8*b^6))*1i)/(8*b^6))*(8*a^4 + 15*b^4 - 20*a^2*b^2))/(8*b^6) + (a*(((225*a^4*b^13)/2 - 300*a^6
*b^11 + 320*a^8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*(64*a*b^17 - 834*a^3*b^15 + 2385*
a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) + (a*(8*a^4 + 15*b^4 - 20*a^2
*b^2)*((tan(c/2 + (d*x)/2)*(128*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15) - (28*a^2*b^16
- 44*a^4*b^14 + 16*a^6*b^12)/b^14 + (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15)
)*(8*a^4 + 15*b^4 - 20*a^2*b^2)*1i)/(8*b^6))*1i)/(8*b^6))*(8*a^4 + 15*b^4 - 20*a^2*b^2))/(8*b^6))/((32*a^16 -
120*a^2*b^14 + 655*a^4*b^12 - 1549*a^6*b^10 + 2069*a^8*b^8 - 1695*a^10*b^6 + 856*a^12*b^4 - 248*a^14*b^2)/b^14
+ (tan(c/2 + (d*x)/2)*(128*a^17 - 450*a^3*b^14 + 2550*a^5*b^12 - 6230*a^7*b^10 + 8530*a^9*b^8 - 7088*a^11*b^6
+ 3584*a^13*b^4 - 1024*a^15*b^2))/b^15 + (a*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9 - 160*a^10*b^7 +
32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b
^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) + (a*(8*a^4 + 15*b^4 - 20*a^2*b^2)*((28*a^2*b^16 - 44*a^4*b^14 + 1
6*a^6*b^12)/b^14 - (tan(c/2 + (d*x)/2)*(128*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15) + (
a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15))*(8*a^4 + 15*b^4 - 20*a^2*b^2)*1i)/(
8*b^6))*1i)/(8*b^6))*(8*a^4 + 15*b^4 - 20*a^2*b^2)*1i)/(8*b^6) - (a*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^
8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 316
0*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) + (a*(8*a^4 + 15*b^4 - 20*a^2*b^2)*((tan(c/
2 + (d*x)/2)*(128*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15) - (28*a^2*b^16 - 44*a^4*b^14
+ 16*a^6*b^12)/b^14 + (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15))*(8*a^4 + 15*
b^4 - 20*a^2*b^2)*1i)/(8*b^6))*1i)/(8*b^6))*(8*a^4 + 15*b^4 - 20*a^2*b^2)*1i)/(8*b^6)))*(8*a^4 + 15*b^4 - 20*a
^2*b^2))/(4*b^6*d)